Chaos, Solitons and Fractals 40 (2009) 466–473

www.elsevier.com/locate/chaos

Chaos control and global synchronization of Liu chaoticsystems using linear balanced feedback control

Heng-Hui Chen *

Department of Mechanical Engineering, HsiuPing Institute of Technology, Taichung 412, Taiwan, ROC

Accepted 31 July 2007

Abstract

Liu chaotic systems exhibit two- or four-scroll attractors and are observed in a variety of engineering phenomena,including rigid body motion, brushless DC motor system and so forth. This study applies the Lyapunov stability the-orem to identify the sufficient conditions for the asymptotic stability of the equilibrium points of Liu chaotic systems. Alinear balanced feedback gain control method is then employed to design a controller to achieve the global synchroni-zation of two identical four-scroll Liu chaotic systems. The feasibility and effectiveness of the proposed chaos stabilityand synchronization schemes are verified via numerical simulations.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

Chaos is a highly important form of nonlinear system behavior, and has attracted considerable attention over thepast two decades [1,2]. Chaos is of fundamental concern in a wide range of fields, including secure communications,optics, chemical and biological systems, and so forth. The desirability, or otherwise, of chaos depends on the particularapplication. Therefore, it is necessary that the chaotic response of a system can be controlled, e.g. by driving the chaoticattractors to a specific region of the system or by eliminating chaos entirely through the application of suitable controllaws. Many researchers have proposed chaos control and synchronization schemes in recent years, including the OGYmethod [3], the PC method [4], linear feedback control [5], adaptive control [6,7], backstepping design [8], active control[9], nonlinear control [10,11] and linear feedback control with LMI [12].

In 2003, Liu and Chen [13] presented a new class of chaotic system similar to the Lorenz and Rossler system, butcharacterized by either two- or four-scroll attractors. This chaos model has attracted considerable interest from manyresearchers. For example, Yassen [5] used the Lyapunov stability theorem to derive the sufficient conditions for the syn-chronization of two identical four-scroll Liu chaotic systems and then applied the Routh–Hurwitz criterion to deter-mine the asymptotic stability conditions which constrained the chaos to unstable equilibrium points in the system.

The current study employs a linear feedback control method to derive the sufficient global stability criteria for con-trolling and synchronizing Liu chaotic systems. The major aim of the study is to establish the criteria which suppress thechaotic reaction of such systems to their unstable equilibrium points. The study commences by using the Lyapunov

0960-0779/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2007.07.098

* Tel./fax: +886 4 24961108.E-mail address: richard@mail.hit.edu.tw

H.-H. Chen / Chaos, Solitons and Fractals 40 (2009) 466–473 467

stability theorem to identify the sufficient conditions for the asymptotic stability of the equilibrium points of the con-trolled system. The Lyapunov stability theorem and a linear balanced feedback gain scheme are then applied to developa feedback control method to synchronize the chaotic response of two identical four-scroll Liu chaotic systems. Thefeasibility of the proposed chaos suppression and synchronization schemes is demonstrated via numerical simulations.

2. System description

Consider the following three-dimensional continuous autonomous system:

_x ¼ axþ d1yz;

_y ¼ by þ d2xz;

_z ¼ czþ d3xy;

ð1Þ

where a, b, c and di are constants. In the literature, it is shown that for a limited set of di, e.g. (d1,d2,d3) = (�1,1,1),(1,�1,�1), (�1,1,1/3), this system exhibits various chaotic attractors at specific values of parameters a, b and c [11,13].

This study considers the condition d1I1 + d2I2 + d3I3 = 0, where I1, I2 and I3 are positive constants. In rigid bodydynamic systems, I1, I2 and I3 represent the principal moments of inertia of the rigid body. According to Liu and Chen[13], the necessary conditions for the system given in (1) to exhibit chaos are b < 0, c < 0,0 < a < � (b + c) and d1 < 0,d2 > 0 and d3 > 0 or d1 > 0, d2 < 0 and d3 > 0, respectively. Without loss of generality, it can be assumed that d1 < 0,d2 > 0 and d3 > 0. The system described in (1) has five equilibrium points, i.e. S0(0,0,0), S1ð�x;�y;�zÞ, S2ð��x;��y;�zÞ,S3ð��x; �y;��zÞ and S4ð�x;��y;��zÞ when ð�x; �y;�zÞ is a real equilibrium point, where �x ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibc=ðd2d3Þ

p;�y ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffica=ðd3d1Þ

pand

�z ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiab=ðd1d2Þ

p; but has only the trivial equilibrium point S0(0,0,0) otherwise.

Depending on the particular choice of system parameters, the chaotic system in (1) may display either 2-scroll or4-scroll chaotic attractors. Under such conditions, it is readily verified that the five equilibrium points of the systemare unstable. Obviously, the eigenvalues of the linearized system at the trivial equilibrium point are k1 = a > 0,k2 = b < 0 and k3 = c < 0, thus S0 is unstable. If a is negative, the origin is a hyperbolic sink.

The Jacobian matrix of system (1) about a nontrivial fixed point, i.e. Si = (a,b,c), is

Ji ¼a d1c d1b

d2c b d2a

d3b d3a c

264

375: ð2Þ

The characteristic equation of Ji has the form

k3 þ c1k2 þ c2kþ c3 ¼ 0; ð3Þ

where c1 = �(a + b + c) > 0, c2 = ab + bc + ca � d2d3a2 � d3d1b

2 � d1d2c2 = 0 and c3 = �abc + ad2d3a

2 + bd3d1b2 +

cd1d2c2 � 2d1d2d3abc = 4abc > 0. The characteristic polynomial in Eq. (3) with nonpositive coefficients is not a Hurwitz

polynomial, i.e. at least one of the determinants D1 = c1, D2 = �c3 and D3 ¼ �c23 < 0 is negative. Hence, according to

the Routh–Hurwitz criterion, all of the nontrivial fixed points are unstable. For any specific set of system parameters, c2

is always equal to zero. Hence, it is certain that all of the nontrivial fixed points are unstable for any choice of systemparameters.

3. Stabilizing the equilibrium points of Liu chaotic systems

Since the equilibrium points of the Liu chaotic system are unstable, a control problem arises. Accordingly, this studydesigns a linear balanced feedback gain control scheme based on the Lyapunov stability theorem to drive the chaoticattractor to a nontrivial fixed point of the system, i.e. Si = (a,b,c), i = 1,2, . . . , 4. It is supposed that the controlled sys-tem has the form

_x ¼ axþ d1yz� k1ðx� aÞ;_y ¼ by þ d2xz� k2ðy � bÞ;_z ¼ czþ d3xy � k3ðz� cÞ;

ð4Þ

where k1, k2 and k3 are positive feedback gains.Let the error states be n1 = x � a, n2 = y � b and n3 = z � c. The error dynamics for the system in (4) are given by

468 H.-H. Chen / Chaos, Solitons and Fractals 40 (2009) 466–473

_n ¼ ðJi � KÞnþ FðnÞ; ð5Þ

where K = diag{k1,k2,k3}, and

A ¼ Ji � K ¼a� k1 d1c d1b

d2c b� k2 d2a

d3b d3a c� k3

264

375; FðnÞ ¼

d1n2n3

d2n1n3

d3n1n2

264

375:

The following Lyapunov function can be constructed

V ¼ nTPn; ð6Þ

where P = diag{I1, I2, I3} is a positive definite matrix which satisfies the condition d1I1 + d2I2 + d3I3 = 0. The derivativeof the Lyapunov function along the trajectory of the system in (5) has the form

_V ¼ _nTPnþ nTP _n ¼ nTðATPþ PAÞnþ ðFTPnþ nTPFÞ ¼ �nTQn; ð7Þ

where FTPn = nTPF = 0, and

Q ¼q1 n1 n2

n1 q2 n3

n2 n3 q3

264

375 ¼

2ðk1 � aÞI1 I3d3c I2d2b

I3d3c 2ðk2 � bÞI2 I1d1a

I2d2b I1d1a 2ðk3 � cÞI3

264

375:

Theorem 1. If the gain matrix K is properly chosen such that Q is a positive definite matrix, then the origin of the error

dynamics system in (5) is globally asymptotically stable, i.e. the chaos is confined to one unstable equilibrium point of the

system in (1).

Proof. By Sylvester’s theorem, all principal minors of Q are strictly positive, i.e.

D1 ¼ q1 ¼ m1 > 0;

D2 ¼ q1q2 � n21 ¼ m2 > 0;

D3 ¼ ðq1q2 � n21Þq3 � q1n2

3 � q2n22 þ 2n1n2n3 ¼ m3 > 0:

Hence, the following conditions can be obtained for the coupling parameters

k1 ¼ aþ m1=ð2I1Þ > maxfa; 0g;k2 ¼ bþ ðm2 þ n2

1Þ=ð2m1I2Þ > maxfb; 0g;k3 ¼ cþ ½m1m3 þ ðm1n3 � n1n2Þ2 þ m2n2

2�=ð2m1m2I3Þ > maxfc; 0g;ð8Þ

where n1 = I3d3c, n2 = I2d2b and n3 = I1d1a. Since n21; n

22 and n2

3 and n1n2n3 have the same values for all of the nontrivialfixed points in system (1), i.e. Si = (a,b,c), i = 1,2, . . . , 4, the same control gains are obtained for all four unstable fixedpoints for any given Ii and mi, i = 1,2,3. h

3.1. Numerical results

To verify the control method described above for stabilizing the equilibrium points, this section chooses a Liu modelwith a four-scroll chaotic attractor for illustration purposes. The system equation is as follows _x ¼ ax� yz; _y ¼ by þ xz,and _z ¼ czþ xy, where the parameters have values of a = 0.5, b = �10 and c = �4. Fig. 1 confirms that this system ischaracterized by a four-scroll chaotic attractor and shows that the system has five unstable equilibrium points, i.e.S0(0,0,0), S1ð�x;�y;�zÞ, S2ð��x;��y;�zÞ, S3ð��x; �y;��zÞ and S4ð�x;��y;��zÞ, where ð�x;�y;�zÞ ¼ ð6:32; 1:41; 2:24Þ.

Assuming that I1 = 1, I2 = 0.5 and I3 = 0.5 and that m1 = 9, m2 = 88 and m3 = 1, the feedback gains required to drivethe chaos to point S1 are found from Eq. (8) to be k1 = 5, k2 = 0 and k3 = 0.272. The controlled system in (4) is integratednumerically using the fourth-order Runge–Kutta method with a time step of 0.005 and an initial state of (x0,y0,z0)1

= (30,30,30). Fig. 2 shows that the controller stabilizes the four-scroll chaotic attractor at unstable equilibrium pointS1 at time t = 8. The state errors are found to be (n1,n2,n3)1 = (�0.37,�0.113,�0.093) · 10�7. Since the system character-istics are symmetrical about the coordinate axes, the results obtained when driving the chaos to points Si, i = 2,3,4 frominitial states of (x0,y0,z0)2 = (�30,�30,30), (x0,y0,z0)3 = (�30,30,�30) and (x0,y0,z0)4 = (30,�30,�30), respectively,

0 2 4 6 8-40

-20

0

20

40

60

t

ξ 1, ξ2, ξ

3

0 2 4 6 8-40

-20

0

20

40

t

ξ 1

0 2 4 6 8-20

0

20

40

t

ξ 2

0 2 4 6 8-20

0

20

40

t

ξ 3

Fig. 2. Variation of state errors (n1,n2,n3) of controlled system in (5) over time for k1 = 5, k2 = 0 and k3 = 0.272.

-400

40

-400

40-40

0

40

xy

z

-40 -20 0 20 40-40

-20

0

20

40

x

y-40 -20 0 20 40

-40

-20

0

20

40

y

z

-40 -20 0 20 40-40

-20

0

20

40

x

z

Fig. 1. Four-scroll chaotic attractor of Liu chaotic system at a = 0.5, b = �10 and c = �4.

H.-H. Chen / Chaos, Solitons and Fractals 40 (2009) 466–473 469

are similar to those shown in Fig. 2. The corresponding state errors are (n1,n2,n3)2 = (0.37,0.113,�0.093) · 10�7,(n1,n2,n3)3 = (0.37,�0.113,0.093) · 10�7 and (n1,n2,n3)4 = (�0.37,0.113,0.093) · 10�7, respectively.

4. Design of chaos synchronization controller

This section uses the Lyapunov stability theorem and a linear balanced feedback gain scheme to design a controllerto synchronize the chaos behavior of two identical four-scroll Liu systems.

470 H.-H. Chen / Chaos, Solitons and Fractals 40 (2009) 466–473

The drive system is given by

_x1 ¼ ax1 þ d1y1z1;

_y1 ¼ by1 þ d2x1z1;

_z1 ¼ cz1 þ d3x1y1:

ð9Þ

The response system has the form

_x2 ¼ ax2 þ d1y2z2 þ u1;

_y2 ¼ by2 þ d2x2z2 þ u2;

_z2 ¼ cz2 þ d3x2y2 þ u3;

ð10Þ

where u1 = �k1e1, u2 = �k2e2 and u3 = �k3e3, in which k1, k2 and k3 are positive feedback gains and e1 = x2 � x1,e2 = y2 � y1 and e3 = z2 � z1 are the error states. The error dynamics in the controlled system in (10) are given by

_e ¼ ðJ� KÞeþ FðeÞ; ð11Þ

where K = diag{k1,k2,k3}, and

A ¼ J� K ¼a� k1 d1z1 d1y1

d2z1 b� k2 d2x1

d3y1 d3x1 c� k3

264

375; FðeÞ ¼

d1e2e3

d2e1e3

d3e1e2

264

375:

The following Lyapunov function can be constructed:

V ¼ eTPe; ð12Þ

where P = diag{I1, I2, I3} is a positive definite matrix which satisfies the condition d1I1 + d2I2 + d3I3 = 0. The derivativeof the Lyapunov function along the trajectory of the system in (11) has the form

_V ¼ _eTPeþ eTP _e ¼ eTðATPþ PAÞeþ ðFTPeþ eTPFÞ ¼ �eTQe; ð13Þ

where FTPe = eTPF = 0, and

Q ¼q1 n1 n2

n1 q2 n3

n2 n3 q3

264

375 ¼

2ðk1 � aÞI1 I3d3z1 I2d2y1

I3d3z1 2ðk2 � bÞI2 I1d1x1

I2d2y1 I1d1x1 2ðk3 � cÞI3

264

375:

Theorem 2. For a given set of system parameters, the two coupled unified chaotic systems in Eqs. (9) and (10) are globally

synchronized if there exists the condition d1I1 + d2I2 + d3I3 = 0 and a suitable linear feedback control K is chosen such that

the following equations hold:

ðiÞ k1 ¼ aþ m1=ð2I1Þ;ðiiÞ k2 ¼ bþ ðm2 þ U 2

1Þ=ð2m1I2Þ;ðiiiÞ k3 ¼ cþ ½m1m3 þ ðm1U 3 þ U 1U 2Þ2 þ m2U 2

2�=ð2m1m2I3Þ;ð14Þ

where Ii and mi, i = 1,2,3 are positive constants.

Proof. Let U1, U2 and U3 be the upper bounds of the absolute values of variables n1, n2 and n3, respectively. From Eq.(13), it can be shown that

_V ¼ �eTQe 6 �ETME; ð15Þ

where E = [je1j je2j je2j]T and

M ¼2ðk1 � aÞI1 �U 1 �U 2

�U 1 2ðk2 � bÞI2 �U 3

�U 2 �U 3 2ðk3 � cÞI3

264

375:

By Sylvester’s theorem, all principal minors of M are strictly positive, i.e. the following conditions hold:

H.-H. Chen / Chaos, Solitons and Fractals 40 (2009) 466–473 471

k1 ¼ aþ m1=ð2I1Þ;k2 ¼ bþ ðm2 þ U2

1Þ=ð2m1I2Þ;k3 ¼ cþ ½m1m3 þ ðm1U 3 þ U 1U 2Þ2 þ m2U 2

2�=ð2m1m2I3Þ: �

ð16Þ

Remark 1. Let U be the maximum upper bound of the absolute values of variables n1, n2 and n3, i.e. U = max{Ui}i=1,2,3.In accordance with Theorem 2, the following equations are obtained for the state feedback gains

ðiÞ k ¼ aþ m =ð2I Þ;

1 1 1

ðiiÞ k2 ¼ bþ ðm2 þ U 2Þ=ð2m1I2Þ;ðiiiÞ k3 ¼ cþ ½m1m3 þ ðm1U þ U 2Þ2 þ m2U 2�=ð2m1m2I3Þ:

ð17Þ

From Eq. (17) (iii), it is clear that an extreme value of k3(m3) is taken on at a boundary point as m3 approaches zero, i.e.m3 = 2em2I3, e! 0+. The maximum upper bound of the absolute values of variables n1, n2 and n3 is given by U = max{-Ui}i=1,2,3 = ImdmX, where Im = max{Ii}i=1,2,3, dm = max{jdij}i=1,2,3 and X = max{jx1j, jy1j, jz1j}. It is assumed that theparameters of the four-scroll chaotic system are specified as d1 = �1, d2 = d3 = 1, a = 0.5, b = �10, c = �4, andI1 = I0, I2 = I3 = I0/2. Then, dm = 1, Im = I0, and X can be found by solving Eq. (9).

Eq. (17) can be rewritten as

k1 ¼ aþ h;

k2 ¼ r1 þ 1=l;

k3 ¼ r2 þ r3l;

ð18Þ

where h = m1/(2I0), r1 = (b + v), r2 = (c 0 + v), c 0 = c + e, r3 = (X + v)2, v = X2/(2h) and

l ¼ I0=j; j ¼ m2=m1:

In accordance with the balanced feedback gain scheme, the following relations are considered: (i) k1 ffi k3 and (ii)k2 ffi k3.

First, let k2 = k3. From Eq. (18), it is shown that

l ¼ ðr1 � r2Þ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr1 � r2Þ2 þ 4r3

qÞ

� ��ð2r3Þ: ð19Þ

If X is sufficiently large that r3� (b � c 0)2/4, then

l � 1=ffiffiffiffir3

p ¼ 1=ðX þ vÞ: ð20Þ

Second, let k1 = k3. From Eqs. (18) and (20), the following linear balanced feedback gains are obtained with respect tothe extreme point ðm�1;m�2;m�3Þ ¼ I0

ffiffiffi5pþ 1

� �X ; I2

0

ffiffiffi5pþ 2

� �X 2; 0þ

� �

k1 ¼ aþ h ¼ aþffiffiffi5pþ 1

� X=2;

k2 ¼ bþ 2vþ X ¼ bþffiffiffi5pþ 1

� X=2;

k3 ¼ c0 þ 2vþ X ¼ c0 þ ðffiffiffi5pþ 1ÞX=2;

ð21Þ

where h ¼ffiffiffi5pþ 1

� �X=2 and v ¼

ffiffiffi5p� 1

� �X=4.

Thus, the sum of the gains is

k1 þ k2 þ k3 ¼ ðaþ bþ c0Þ þ 3ðffiffiffi5pþ 1ÞX=2: ð22Þ

The values of the feedback gains k1, k2 and k3 are proportional to the maximum upper bound, X, of the absolute valuesof variables x1, y1 and z1. Furthermore, the feedback gains are approximately equal to one another, i.e. they are balanced.

4.1. Numerical results

Fig. 1 shows that the solutions x(t), y(t) and z(t) corresponding to the initial states x(0) = 1, y(0) = 1 and z(0) = 1,respectively, are bounded and satisfy the inequalities �29.7 < x < 28.5, �21 < y < 22.1 and �26.4 < z < 22.2, respec-tively. Hence, the maximum upper bound, X, can be specified as X = 30.

472 H.-H. Chen / Chaos, Solitons and Fractals 40 (2009) 466–473

The simulation results presented in Figs. 3 and 4 verify the effectiveness of the proposed chaos synchronization con-trol scheme. In Fig. 3a, the feedback gains k1, k2 and k3 successfully synchronize the chaos in the two systems as X

varies, but k3 increases considerably compared to k1 and k2 for the case of constant parameters (m1,m2) = (9,88) shown

0 10 20 30

0

500

1000

1500

2000

X

k 1,k

2,k

3

0 10 20 3010

0

101

102

X

m1,m

2

0 10 20 30-20

0

20

40

60

X

k 1,k

2,k

3

0 10 20 3010

-2

100

102

104

X

m1* ,m

2*

a

c d

b

Fig. 3. Variations of feedback gains k1 (—), k2 (––) and k3 (—.) and fixed parameters m1 (—) and m2 (––) with X: (a) feedback gains ascomputed from Eq. (18) at fixed parameters (m1,m2) = (9,88), (b) fixed parameters (m1,m2) = (9,88), (c) balanced feedback gains ascomputed from Eq. (21) at extreme parameters ðm�1;m�2Þ, and (d) extreme parameters ðm�1;m�2Þ.

0 0.5 1 1.5 2 2.5 3-0.5

0

0.5

1

1.5

t

e 1,e

2,e

3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.5

1

1.5

t

e 1,e

2,e

3

a

b

Fig. 4. Synchronization error dynamics of two identical four-scroll Liu chaotic systems with feedback gains computed from Eqs. (18)and (21) at X = 30 and: (a) fixed parameters (m1,m2) = (9,88) and (b) extreme parameters ðm�1;m�2Þ ¼ ð97; 3813Þ.

H.-H. Chen / Chaos, Solitons and Fractals 40 (2009) 466–473 473

in Fig. 3b. Fig. 3c shows that the feedback gains k1, k2 and k3 are approximately equal for the extreme parametersðm�1;m�2Þ shown in Fig. 3d. In general, Fig. 3 shows that for the maximum upper bound of X = 30, the feedback gainsof the controlled system are K1 = (k1,k2,k3) = (5,100,1825) and K2 = (k1,k2,k3) = (49,38.5,44.5), respectively, withcorresponding parameters of (m1,m2) = (9,88) and ðm�1;m�2Þ ¼ ð97; 3813Þ. The initial states of the drive and responsesystems are x1(0) = 1, y1(0) = 1 and z1(0) = 1 and x2(0) = 2.5, y2(0) = 2.5 and z2(0) = 2.5, respectively. Fig. 4 presentsthe simulation results obtained for the two identical four-scroll systems. It can be seen that the proposed linear balancedfeedback gain control scheme successfully achieves a rapid chaos synchronization of the two systems.

5. Conclusion

This study has investigated the problem of chaos control and synchronization in Liu chaotic systems. The sufficientconditions for the stability of the equilibrium points of the controlled system have been obtained using the Lyapunovstability theorem. Additionally, appropriate linear balanced feedback gains have been derived to ensure the global syn-chronization of two identical four-scroll chaotic systems. The feasibility and effectiveness of the chaos suppression andsynchronization schemes have been verified via numerical simulations.

Acknowledgement

This study was supported by the National Science Council, Republic of China, under grant number NSC 93-2218-E-164-001.

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